Ngā Tauira Pātai e Matapaki ana i ngā Pūnaha o ngā Whārite Raina me ngā Taurite Kore
He kaupapa nui ngā pūnaha whārite rārangi me ngā taurite kore i roto i te pāngarau, ā, he whānui te whakamahinga i roto i ngā momo mara, pērā i te ōhanga, te pūtaiao, me te hangarau. I roto i tēnei tuhinga, ka matapakihia e mātou ngā tauira rapanga e pā ana ki ngā pūnaha whārite rārangi me ngā taurite kore, me pēhea hoki te whakaoti taipitopito i aua rapanga.
Te Whakamāramatanga o tētahi Pūnaha Whārite Raina
Ko tētahi pūnaha whārite rārangi he mea tito mai i ngā whārite rārangi e rua, neke atu rānei, e hono ana tetahi ki tetahi. Ko ētahi tauira:
\[
\begin{ngā take}
2x + 3y = 5
4x – y = 1
\end{ngā take}
\]
Ko te whāinga o te whakaoti i tēnei pūnaha he kimi i ngā uara o \(x\) me \(y\) e tutuki ana i ngā whārite e rua i te wā kotahi.
Ngā Tikanga mō te Whakaoti i ngā Pūnaha o ngā Whārite Raina
He maha ngā tikanga hei whakaoti rapanga i ngā pūnaha whārite rārangi, tae atu ki:
1. Tikanga Whakakapinga
2. Tikanga Whakakore
3. Tikanga Matrix (Whakamuri, Gauss-Jordan rānei)
Tauira Pātai 1: Tikanga Whakakapinga
Me whakaoti tātou i te pūnaha e whai ake nei mā te whakamahi i te tikanga whakakapinga:
\[
\begin{ngā take}
x + 2y = 10
3x – y = 5
\end{ngā take}
\]
Ngā Hipanga:
1. Wehea tētahi o ngā taurangi i roto i tētahi o ngā whārite.
Mai i te whārite tuatahi, ka wehea e tātou a \(x\):
\[
x = 10 – 2y
\]
2. Whakakapia te kīanga kua kitea ki roto i te whārite kē atu.
Tāpirihia te \(x = 10 – 2y\) ki te whārite tuarua:
\[
3(10 – 2y) – y = 5
\]
Whakatauhia mō \(y\):
\[
30 – 6y – y = 5
\]
\[
30 – 7y = 5
\]
\[
-7y = -25
\]
\[
y = \frac{25}{7}
\]
3. Whakamahia ngā uara kua kitea hei kimi i ētahi atu taurangi.
Whakakapia te \(y = \frac{25}{7}\) ki roto i te kīanga mō \(x\):
\[
x = 10 – 2\left(\frac{25}{7}\right)
\]
\[
x = 10 – \frac{50}{7}
\]
\[
x = \frac{70}{7} – \frac{50}{7}
\]
\[
x = \frac{20}{7}
\]
Nō reira, ko ngā otinga mō te pūnaha ko \( x = \frac{20}{7} \) me \( y = \frac{25}{7} \).
Tauira Pātai 2: Tikanga Whakakore
Muri iho, me whakamahi tātou i te tikanga whakakore hei whakaoti i te pūnaha e whai ake nei:
\[
\begin{ngā take}
2x + 3y = 12
4x + 6y = 24
\end{ngā take}
\]
I tēnei tauira, ka kite tātou ko te whārite tuarua he tauwehenga o te whārite tuatahi. Hei whakaiti i te pūnaha, ka taea e tātou te whakarea i te whārite tuatahi ki te 2, kātahi ka tangohia mai i te whārite tuarua:
1. Whakareatia te whārite tuatahi ki te 2:
\[
2(2x + 3y) = 2 \cdot 12
\]
\[
4x + 6y = 24
\]
2. Tangohia te whārite tuatahi kua whakareatia mai i te whārite tuarua:
\[
(4x + 6y) – (4x + 6y) = 24 – 24
\]
\[
0 = 0
\]
Ka puta ko te \(0 = 0\), e tohu ana he mutunga kore ngā otinga a te pūnaha, ā, he mea whakawhirinaki ēnei whārite.
Tauira Pātai 3: Ngā Tauritekore Rārangi
He rite ngā mātāpono o ngā taurite kore rārangi ki ngā whārite rārangi, engari kei roto ko ngā tohu taurite kore pērā i te \(<, \leq, >, \geq\). Me titiro tātou ki tētahi tauira māmā:
\[
\begin{ngā take}
3x – y < 7 \\ 2x + y \geq 4 \end{cases} \] Ngā Hipanga: 1. Ka whakamahia e mātou te tikanga whakairoiro hei whakatau i te rohe otinga o tēnei pūnaha. Tuhia ki te kauwhata ia taurite kore. 2. Tahurihia te taurite kore ki tētahi whārite hei whakatau i te rārangi rohe: Mō \(3x - y < 7\), ko te rārangi rohe ko \(3x - y = 7\)
